Specifically for integrable partial differential systems which are *dispersionless a.k.a. hydrodynamic-type*, i.e. can be written as quasilinear first-order homogeneous systems, in the case of *two* independent variables see these lecture notes, Section 3 of this article, and references therein; in the case of *three*independent variables see Section 3 of this same article, subsubsection 10.3.3 of the Dunajski book mentioned by @just-learning, and these lecture notes, and references therein; for the case of *four* independent variables see introductory part of this article, the Dunajski book again, and references therein.